Atlas › Modular forms › Weight 2 › Level 875 › 875.2.a

Newspace 875.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(875))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
875.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - q^{6} - 2q^{7} + O(q^{8})$	+
875.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{3} + 6q^{4} + 5q^{6} + 2q^{7} + O(q^{8})$	−
875.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{3} + 6q^{4} + 5q^{6} - 2q^{7} + O(q^{8})$	+
875.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{7} + O(q^{8})$	−
875.2.a.e	6	degree 6	$6q - 2q^{2} - 5q^{3} + 4q^{4} - 2q^{6} + 6q^{7} + O(q^{8})$	−
875.2.a.f	6	degree 6	$6q - 2q^{2} - 3q^{3} + 4q^{4} - 2q^{6} - 6q^{7} + O(q^{8})$	+
875.2.a.g	6	degree 6	$6q + 2q^{2} + 3q^{3} + 4q^{4} - 2q^{6} + 6q^{7} + O(q^{8})$	−
875.2.a.h	6	degree 6	$6q + 2q^{2} + 5q^{3} + 4q^{4} - 2q^{6} - 6q^{7} + O(q^{8})$	+
875.2.a.i	8	degree 8	$8q - q^{2} + 8q^{3} + 13q^{4} + 2q^{6} + 8q^{7} + O(q^{8})$	−
875.2.a.j	8	degree 8	$8q + q^{2} - 8q^{3} + 13q^{4} + 2q^{6} - 8q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
875.2.a.a	$x^{2} - x - 1$
875.2.a.b	$x^{2} + 3x - 9$
875.2.a.c	$x^{2} - 3x - 9$
875.2.a.d	$x^{2} + x - 1$
875.2.a.e	$x^{6} + 7x^{5} + 6x^{4} - 34x^{3} - 49x^{2} + 35x + 59$
875.2.a.f	$x^{6} + 5x^{5} - 4x^{4} - 22x^{3} + 29x^{2} - 11x + 1$
875.2.a.g	$x^{6} - 5x^{5} - 4x^{4} + 22x^{3} + 29x^{2} + 11x + 1$
875.2.a.h	$x^{6} - 7x^{5} + 6x^{4} + 34x^{3} - 49x^{2} - 35x + 59$
875.2.a.i	$x^{8} - 7x^{7} - 13x^{6} + 155x^{5} - 104x^{4} - 625x^{3} + 543x^{2} + 274x - 199$
875.2.a.j	$x^{8} + 7x^{7} - 13x^{6} - 155x^{5} - 104x^{4} + 625x^{3} + 543x^{2} - 274x - 199$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(875))$	48
Old forms	43
Cusp forms $S_{2}(\Gamma_0(875))$	91
Eisenstein series $E_{2}(\Gamma_0(875))$	19
Modular forms $M_{2}(\Gamma_0(875))$	110

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(875)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(175))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 5, 7, 25, 35, 125, 175, 875]: [0, 1, 1, 3, 6, 13, 22, 102] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(875, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 875:2 1000

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